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Summation-by-parts (SBP) operators are popular building blocks for systematically developing stable and high-order accurate numerical methods for time-dependent differential equations. The main idea behind existing SBP operators is that the solution is assumed to be well approximated by polynomials up to a certain degree, and the SBP operator should therefore be exact for them. However, polynomials might not provide the best approximation for some problems, and other approximation spaces may be more appropriate. In this paper, a theory for SBP operators based on general function spaces is developed. We demonstrate that most of the established results for polynomial-based SBP operators carry over to this general class of SBP operators. Our findings imply that the concept of SBP operators can be applied to a significantly larger class of methods than currently known. We exemplify the general theory by considering trigonometric, exponential, and radial basis functions., 22 pages, 6 figures

Finite-difference based approaches are common for approximating the Caputo fractional derivative. Often, these methods lead to a reduction of the convergence rate that depends on the fractional order. In this note, we approximate the expressions in the fractional derivative components using a separate quadrature rule for the integral and a separate discretization of the derivative in the integrand. By this approach, the error terms from the Newton–Cotes quadrature and the differentiation are isolated and it is possible to conclude that the order dependent error is inevitable when the end points of the interval are included in the quadrature. Furthermore, we show experimentally that the theoretical findings carries over to quadrature rules without the end points included. Finally we show how to increase accuracy for smooth functions, and compensate for the order dependent loss. Funding: BAJ acknowledges support from the National Research Foundation of South Africa under Grant Numbers 129119 and 127567. FL and JN were supported by Vetenskapsrådet, Sweden Grant Numbers 2018-05084 and 2021-05484.

A technique for coupling an intrusive and non-intrusive uncertainty quantification method is proposed. The intrusive approach uses a combination of polynomial chaos and stochastic Galerkin projection. The non-intrusive method uses numerical integration by combining quadrature rules and the probability density functions of the prescribed uncertainties. A stable coupling procedure between the two methods at an interface is constructed. The efficiency of the hybrid method is exemplified using hyperbolic systems of equations, and verified by numerical experiments.

In the hyperbolic research community, there exists the strong belief that a continuous Galerkin scheme is notoriously unstable and additional stabilization terms have to be added to guarantee stability. In the first part of the series [6], the application of simultaneous approximation terms for linear problems is investigated where the boundary conditions are imposed weakly. By applying this technique, the authors demonstrate that a pure continuous Galerkin scheme is indeed linearly stable if the boundary conditions are imposed in the correct way. In this work, we extend this investigation to the nonlinear case and focus on entropy conservation. By switching to entropy variables, we provide an estimation of the boundary operators also for nonlinear problems, that guarantee conservation. In numerical simulations, we verify our theoretical analysis.

We derive boundary conditions and estimates based on the energy and entropy analysis of systems of the nonlinear shallow water equations in two spatial dimensions. It is shown that the energy method provides more details, but is fully consistent with the entropy analysis. The details brought forward by the nonlinear energy analysis allow us to pinpoint where the difference between the linear and nonlinear analysis originate. We find that the result from the linear analysis does not necessarily hold in the nonlinear case. The nonlinear analysis leads in general to a different minimal number of boundary conditions compared with the linear analysis. In particular, and contrary to the linear case, the magnitude of the flow does not influence the number of required boundary conditions. Funding: Vetenskapsradet, Sweden [2018-05084 VR, 2020-03642 VR]

In the context of coupling hyperbolic problems, the maximum stable time step of an explicit numerical scheme may depend on the design of the coupling procedure. If this is the case, the coupling procedure is sensitive to changes in model parameters independent of the Courant-Friedrichs-Levy condition. This sensitivity can cause artificial stiffness that degrades the performance of a numerical scheme. To overcome this problem, we present a systematic and general procedure for weakly imposing coupling conditions via penalty terms in a provably non-stiff manner. The procedure can be used to construct both energy conservative and dissipative couplings, and the user is given control over the amount of dissipation desired. The resulting formulation is simple to implement and dual consistent. The penalty coefficients take the form of projection matrices based on the coupling conditions. Numerical experiments demonstrate that this procedure results in both optimal spectral radii and superconvergent linear functionals. Funding Agencies|Linkoping University; Southern California Earthquake Center [10148]; National Science FoundationNational Science Foundation (NSF) [EAR-1600087]; United States Geological SurveyUnited States Geological Survey [G17AC00047]; NSF-ACI Award [1450451]; Vetenskapsradet, SwedenSwedish Research Council [2018-05084 VR]

We combine existing summation-by-parts discretization methods to obtain a simplified numerical framework for partial differential equations posed on complex multi-block/element domains. The interfaces (conforming or non-conforming) between blocks are treated with inner-product-preserving interpolation operators, and the result is a high-order multi-block operator on summation-by-parts form that encapsulates both the metric terms as well as the interface treatments. This enables for a compact description of the numerical scheme that mimics the essential features of its continuous counterpart. Furthermore, the stability analysis on a multi-block domain is simplified for both for linear and nonlinear equations, since no problem-specific interface conditions need to be derived and implemented. We exemplify the combined operator technique by considering a nonlinearly stable discrete formulation of the incompressible Navier-Stokes equations and perform calculations on an underlying multi-block domain. Funding: Vetenskapsradet, Sweden [2020-03642, 2018-05084, 2021-05484]; Swedish e-Science Research Center (SeRC)

We show that a specific skew-symmetric form of nonlinear hyperbolic problems leads to energy and entropy bounds. Next, we exemplify by considering the compressible Euler equations in primitive variables, transform them to skew-symmetric form and show how to obtain energy and entropy estimates. Finally we show that the skew-symmetric formulation lead to energy and entropy stable discrete approximations if the scheme is formulated on summation-by-parts form. Funding agencies: Vetenskapsradet, Sweden [2018-05084 VR]; Swedish e-Science Research Center (SeRC)

We use the energy method to study the well-posedness of initial-boundary value problems approximated by overset mesh methods in one and two space dimensions for linear constant-coefficient hyperbolic systems. We show that in one space dimension, for both scalar equations and systems of equations, the problem where one domain partially oversets another is well-posed when characteristic coupling conditions are used. If a system cannot be diagonalized, as is usually the case in multiple space dimensions, then the energy method does not give proper bounds in terms of initial and boundary data. For those problems, we propose a novel penalty approach. We show, by using a global energy that accounts for the energy in the overlap region of the domains, that under well-defined conditions on the coupling matrices the penalized overset domain problems are energy bounded, conservative, well-posed and have solutions equivalent to the original single domain problem. Funding: Simons Foundation [426393]; Vetenskapsradet, SwedenSwedish Research Council [2018-05084 VR]; Swedish e-Science Research Center (SeRC); Klaus-Tschira Stiftung; European Research CouncilEuropean Research Council (ERC)European Commission [71448]

Energy stable and conservative nonlinear weakly imposed interface conditions for the incompressible Euler equations are derived in the continuous setting. By discretely mimicking the continuous analysis using summation-by-parts operators, we prove that the numerical scheme is stable and conservative. The theoretical findings are verified by numerical experiments. Funding: Vetenskapradet [2018-05084, 2021-05484]

Close to solid walls, at high Reynolds numbers, fluids may develop steep gradients which require a fine mesh for an accurate simulation of the turbulent boundary layer. An often used cure is to use a wall model instead of a fine mesh, with the drawback that modeling is introduced, leading to possibly unstable numerical schemes. In this paper, we leave the modeling aside, take it for granted, and propose a new set of provably energy stable boundary procedures for the incompressible Navier-Stokes equations. We show that these new boundary procedures lead to numerical results with high accuracy even for coarse meshes where data is partially obtained from a wall model. Funding: Vetenskapsradet, SwedenSwedish Research Council [2018-05084 VR]; Swedish e-Science Research Center (SeRC)

Numerical solvers of initial boundary value problems will exhibit instabilities and loss of accuracy unless carefully designed. The key property that leads to convergence is stability, which this t ...

We prove that the most common filtering procedure for nodal discontinuous Galerkin (DG) methods is stable. The proof exploits that the DG approximation is constructed from polynomial basis functions and that integrals are approximated with high-order accurate Legendre–Gauss–Lobatto quadrature. The theoretical discussion re-contextualizes stable filtering results for finite difference methods into the DG setting. Numerical tests verify and validate the underlying theoretical results.

We extend the convergence results in Svärd and Nordström (2019) [7] for single-domain energy-stable high-order finite difference schemes, to include domains split into several grid blocks. The analysis also demonstrates that reflective boundary conditions enjoy the same convergence properties. Finally, we briefly indicate that these results (and the previous ones in [7]) also hold in multiple dimensions. publishedVersion

We consider accurate and stable interpolation procedures for numerical simulations utilizingtime dependent adaptive meshes. The interpolation of numerical solution valuesbetween meshes is considered as a transmission problem with respect to the underlying semidiscretizedequations, and a theoretical framework using inner product preserving operatorsis developed, which allows for both explicit and implicit implementations. The theory issupplemented with numerical experiments demonstrating practical benefits of the new stableframework. For this purpose, new interpolation operators have been designed to be used withmulti-block finite difference schemes involving non-collocated, moving interfaces. Funding:National Research Foundation of South AfricaNational Research Foundation - South Africa [89916]; Vetenskapsradet, SwedenSwedish Research Council [2018-05084_VR]

The influence of different boundary conditions on the spectral properties of the incompressible Navier-Stokes equations is investigated. By using the Fourier-Laplace transform technique, we determine the spectra, extract the decay rate in time, and investigate the dispersion relation. In contrast to an infinite domain, where only diffusion affects the convergence, we show that also the propagation speed influence the rate of convergence to steady state for a bounded domain. Once the continuous equations are analyzed, we discretize using high-order finite-difference operators on summation-by-parts form and demonstrate that the continuous analysis carries over to the discrete setting. The theoretical results are verified by numerical experiments, where we highlight the necessity of high accuracy for a correct description of time-dependent phenomena. Funding agency: The Swedish e-Science Research Centre (SeRC)

We use the behavior of the $$L_{2}$$ L 2 norm of the solutions of linear hyperbolic equations with discontinuous coefficient matrices as a surrogate to infer stability of discontinuous Galerkin spectral element methods (DGSEM). Although the $$L_{2}$$ L 2 norm is not bounded in terms of the initial data for homogeneous and dissipative boundary conditions for such systems, the $$L_{2}$$ L 2 norm is easier to work with than a norm that discounts growth due to the discontinuities. We show that the DGSEM with an upwind numerical flux that satisfies the Rankine–Hugoniot (or conservation) condition has the same energy bound as the partial differential equation does in the $$L_{2}$$ L 2 norm, plus an added dissipation that depends on how much the approximate solution fails to satisfy the Rankine–Hugoniot jump.

We develop a novel numerical scheme for the simulation of dissipative quantum dynamics following from two-body Lindblad master equations. All defining continuum properties of the Lindblad dynamics, hermiticity, positivity and in particular trace conservation of the evolved density matrix are preserved. The central ingredient is a new spatial difference operator, which not only fulfils the summation by parts (SBP) property but also implements a continuum reparametrization property. Using the time evolution of a heavy-quark anti-quark bound state in a hot thermal medium as an explicit example, we show how the reparametrization neutral summation-by-parts (RN-SBP) operator preserves the continuum properties of the theory., 34 pages, 7 figures, open-access code available via https://doi.org/10.5281/zenodo.3744460

Total variation diminishing multigrid methods have been developed for first order accurate discretizations of hyperbolic conservation laws. This technique is based on a so-called upwind biased residual interpolation and allows for algorithms devoid of spurious numerical oscillations in the transient phase. In this paper, we justify the introduction of such prolongation and restriction operators by rewriting the algorithm in a matrix-vector notation. This perspective sheds new light on multigrid procedures for hyperbolic problems and provides a direct extension for high order accurate difference approximations. The new multigrid procedure is presented, advantages and disadvantages are discussed and numerical experiments are performed. Funding agencies: Linkoping University; VINNOVA, the Swedish Governmental Agency for Innovation SystemsVinnova [2013-01209]

Linearisation is often used as a first step in the analysis of nonlinear initial boundary value problems. The linearisation procedure frequently results in a confusing contradiction where the nonlinear problem conserves energy and has an energy bound but the linearised version does not (or vice versa). In this paper we attempt to resolve that contradiction and relate nonlinear energy conserving and bounded initial boundary value problems to their linearised versions and the related dual problems. We start by showing that a specific skew-symmetric form of the primal nonlinear problem leads to energy conservation and a bound. Next, we show that this specific form together with a non-standard linearisation procedure preserves these properties for the new slightly modified linearised problem. We proceed to show that the corresponding linear and nonlinear dual (or self-adjoint) problems also have bounds and conserve energy due to this specific formulation. Next, the implication of the new formulation on the choice of boundary conditions is discussed. A straightforward nonlinear and linear analysis may lead to a different number and type of boundary conditions required for an energy bound. We show that the new formulation sheds somelight on this contradiction. We conclude by illustrating that the new continuous formulation automatically leads toenergy stable and energy conserving numerical approximations for both linear and nonlinear primal and dual problems if the approximations are formulated on summation-by-parts form. Funding: Vetenskapsradet, SwedenSwedish Research Council [2018-05084 VR]; Swedish e-Science Research Center (SeRC)

We study an incompletely parabolic system in three space dimensions with stochastic boundary and initial data. We show how the variance of the solution can be manipulated by the boundary conditions, while keeping the mean value of the solution unaffected. Estimates of the variance of the solution is presented both analytically and numerically. We exemplify the technique by applying it to an incompletely parabolic model problem, as well as the one-dimensional compressible Navier–Stokes equations. Funding agencies: European Commission [ACP3-GA-2013-605036]

We introduce the notion of a transmission problem to describe a general class of problems where different dynamics are coupled in time. Well-posedness and stability are analysed for continuous and discrete problems using both strong and weak formulations, and a general transmission condition is obtained. The theory is applied to the coupling of fluid-acoustic models, multi-grid implementations, adaptive mesh refinements, multi-block formulations and numerical filtering. Funding agencies: Swedish Meteorological and Hydrological Institute (SMHI)

Multigrid schemes for high order finite difference methods on summation-by-parts form are studied by comparing the effect of different interpolation operators. By using the standard linear prolongation and restriction operators, the Galerkin condition leads to inaccurate coarse grid discretizations. In this paper, an alternative class of interpolation operators that bypass this issue and preserve the summation-by-parts property on each grid level is considered. Clear improvements of the convergence rate for relevant model problems are achieved. Funding agencies: VINNOVA, the Swedish Governmental Agency for Innovation Systems [2013-01209]

Temporal error bounds for the wave equation expressed on second order form are investigated. We show that, with appropriate choices of boundary conditions, the time and space derivatives of the error are bounded even for long times. No long time bound on the error itself is obtained, although numerical experiments indicate that a bound exists.

A one-dimensional steady-state advection-diffusion problem using summation-by-parts operators is investigated. For approximating the second derivative, a wide stencil is used, which simplifies implementation and stability proofs. However, it also introduces spurious, oscillating, modes for all mesh sizes. We prove that the size of the spurious modes is equal to the size of the truncation error for a stable approximation and hence disappears with the convergence rate. The theoretical results are verified with numerical experiments. Funding agencies:This project was funded by the Swedishe-science Research Center (SeRC). Thefunding source had no involvement in thestudy design, collection and analysis ofdata, or in writing and submitting thisarticle

We consider the coupling of parabolic problems discretized using difference operators on summation-by-parts (SBP) form with interface conditions imposed weakly. In [1, 2], it was shown that conserv ...

In this paper we study boundary conditions for linear hyperbolic systems of equations and the corresponding dual problem. In particular, we show that the primal and dual boundary conditions are related by a simple scaling relation. It is also shown that the weak dual problem can be derived directly from the weak primal problem. Based on the continuous analysis, we discretize and perform computations with a high-order finite difference scheme on summation- by-parts form with weak boundary conditions. It is shown that the results obtained in the continuous analysis lead directly to stability results for the primal and dual discrete problems. Numerical experiments corroborate the theoretical results.

Diagonal norm finite difference based time integration methods in summation-by-parts form are investigated. The second, fourth, and sixth order accurate discretizations are proven to have eigenvalues with strictly positive real parts. This leads to provably invertible fully discrete approximations of initial boundary value problems. Our findings also allow us to conclude that the Runge--Kutta methods based on second, fourth, and sixth order summation-by-parts finite difference time discretizations automatically satisfy previously unreported stability properties. The procedure outlined in this article can be extended to even higher order summation-by-parts approximations with repeating stencil. Funding agencies: VINNOVA, the Swedish Governmental Agency for Innovation SystemsVinnova [2013-01209]

Both the energy method and the Laplace transform method are frequently used for determining the number of boundary conditions required for a well posed initial boundary value problem. We show that these two distinctly different methods yield the same results. The continuous energy method can be mimicked exactly in the corresponding semidiscrete problems discretized using the summation-by-parts technique. Hence the analysis of well posedness and stability can bypass the more unwieldy Laplace transform method. Funding agencies: The first author was supported by Vetenskapsrådet, Sweden grant 2018-05084 VR. The second author was supported by National Science Foundation grant DMS-2012296.

High frequency errors are always present in numerical simulations since no difference stencil is accurate in the vicinity of the $$\pi $$π-mode. To remove the defective high wave number information from the solution, artificial dissipation operators or filter operators may be applied. Since stability is our main concern, we are interested in schemes on summation-by-parts (SBP) form with weak imposition of boundary conditions. Artificial dissipation operators preserving the accuracy and energy stability of SBP schemes are available. However, for filtering procedures it was recently shown that stability problems may occur, even for originally energy stable (in the absence of filtering) SBP based schemes. More precisely, it was shown that even the sharpest possible energy bound becomes very weak as the number of filtrations grow. This suggest that successful filtering include a delicate balance between the need to remove high frequency oscillations (filter often) and the need to avoid possible growth (filter seldom). We will discuss this problem and propose a remedy.

In the hyperbolic community, discontinuous Galerkin approaches are mainly applied when finite element methods are considered. As the name suggested, the DG framework allows a discontinuity at the element interfaces, which seems for many researchers a favorable property in case of hyperbolic balance laws. On the contrary, continuous Galerkin method obtained from a straightforward discretisation of the weak form of the PDEs appear to be unsuitable for hyperbolic problems. To remedy this issue, stabilization terms are usually added and various formulations can be found in the literature. There exists still the perception that continuous Galerkin methods are not suited to hyperbolic problems, and the reason of this is the continuity of the approximation. However, this perception is not true and the stabilization terms can be removed, in general, provided the boundary conditions are suitable. In this paper, we deal with this problem, and present a different approach. We use the boundary conditions to stabilize the scheme following a procedure that are frequently used in the finite difference community. Here, the main idea is to impose the boundary conditions weakly and specific boundary operators are constructed such that they guarantee stability. This approach has already been used in the DG framework, but here we apply it with a continuous Galerkin scheme. No internal dissipation is needed even if unstructured grids are used. Further, we point out that we do not need exact integration, it suffices if the quadrature rule and the norm in the differential operator are the same, such that the summation-by-parts (SBP) property is fulfilled meaning that a discrete Gauss Th. is valid. This contradicts the perception in the hyperbolic community that stability issues for pure Galerkin scheme exist. In numerical simulations, we verify our theoretical analysis., 28 pages, 10 figures

Many physical phenomena can be described mathematically by means of partial differential equations. These mathematical formulations are said to be well-posed if a unique solution, bounded by the gi ...

We consider two hyperbolic systems in first order form of different size posed on two domains. Our ambition is to derive general conditions for when the two systems can and cannot be coupled. The adjoint equations are derived and well-posedness of the primal and dual problems is discussed. By applying the energy method, interface conditions for the primal and dual problems are derived such that the continuous problems are well posed. The equations are discretized using a high order finite difference method in summation-by-parts form and the interface conditions are imposed weakly in a stable way, using penalty formulations. It is shown that one specic choice of penalty matrices leads to a dual consistent scheme. By considering an example, it is shown that the correct physical coupling conditions are contained in the set of well posed coupling conditions. It is also shown that dual consistency leads to superconverging functionals and reduced stiffness.

The effect of an inaccurate geometry description on the solution accuracy of a hyperbolic problem is discussed. The inaccurate geometry can for example come from an imperfect CAD system, a faulty mesh generator, bad measurements or simply a misconception. We show that inaccurate geometry descriptions might lead to the wrong wave speeds, a misplacement of the boundary conditions, to the wrong boundary operator and a mismatch of boundary data. The errors caused by an inaccurate geometry description may affect the solution more than the accuracy of the specific discretization techniques used. In extreme cases, the order of accuracy goes to zero. Numerical experiments corroborate the theoretical results.

Unresolved gradients produce numerical oscillations and inaccurate results. The most straightforward solution to such a problem is to increase the resolution of the computational grid. However, this is often prohibitively expensive and may lead to ecessive execution times. By training a neural network to predict the shape of the solution, we show that it is possible to reduce numerical oscillations and increase both accuracy and efficiency. Data from the neural network prediction is imposed using multiple penalty terms inside the domain.

Artificial neural networks together with associated computational libraries provide a powerful framework for constructing both classification and regression algorithms. In this paper we use neural networks to design linear and non-linear discrete differential operators. We show that neural network based operators can be used to construct stable discretizations of initial boundary-value problems by ensuring that the operators satisfy a discrete analogue of integration-by-parts known as summation-by-parts. Our neural network approach with linear activation functions is compared and contrasted with a more traditional linear algebra approach. An application to overlapping grids is explored. The strategy developed in this work opens the door for constructing stable differential operators on general meshes.

n/a Classified in the journal as "Short note"

We review and extend the list of stability and convergence properties satisfied by Runge-Kutta (RK) methods that are associated with Summation-By-Parts (SBP) operators, herein called RK-SBP methods. The analysis covers classical, generalized as well as upwind SBP operators. Previous work on the topic has relied predominantly on energy estimates. In contrast, we derive all results using a purely algebraic approach that rests on the well-established theory of RK methods. The purpose of this paper is to provide a bottom-up overview of stability and convergence results for linear and non-linear problems that relate to general RK-SBP methods. To this end, we focus on the RK viewpoint, since this perspective so far is largely unexplored. This approach allows us to derive all results as simple consequences of the properties of SBP methods combined with well-known results from RK theory. In this way, new proofs of known results such as A-, L- and B-stability are given. Additionally, we establish previously unreported results such as strong S-stability, dissipative stability and stiff accuracy of certain RK-SBP methods. Further, it is shown that a subset of methods are B-convergent for strictly contractive non-linear problems and convergent for non-linear problems that are both contractive and dissipative.

We introduce solution dependent finite difference stencils whose coefficients adapt to the current numerical solution by minimizing the truncation error in the least squares sense. The resulting scheme has the resolution capacity of dispersion relation preserving difference stencils in under-resolved domains, together with the high order convergence rate of conventional central difference methods in well resolved regions. Numerical experiments reveal that the new stencils outperform their conventional counterparts on all grid resolutions from very coarse to very fine.

We derive analytic solutions to the scalar and vector advection equation with variable coefficients in one spatial dimension using Laplace transform methods. These solutions are used to investigate how accuracy and stability are influenced by the presence of discontinuous wave speeds when applying high-order-accurate, skew-symmetric finite difference methods designed for smooth wave speeds. The methods satisfy a summation-by-parts rule with weak enforcement of boundary conditions and formal order of accuracy equal to 2, 3, 4 and 5. We study accuracy, stability and convergence rates for linear wave speeds that are (a) constant, (b) non-constant but smooth, (c) continuous with a discontinuous derivative, and (d) constant with a jump discontinuity. Cases (a) and (b) correspond to smooth wave speeds and yield stable schemes and theoretical convergence rates. Non-smooth wave speeds [cases (c) and (d)], however, reveal reductions in theoretical convergence rates and in the latter case, the presence of an instability.

The coupling of the compressible and incompressible Navier-Stokes equations is considered. Our ambition is to take a first step towards a provably well posed and stable coupling procedure. We study a simplified setting with a stationary planar interface and small disturbances from a steady background flow with zero velocity normal to the interface. The simplified setting motivates the use of the linearized equations, and we derive interface conditions such that the continuous problem satisfy an energy estimate. The interface conditions can be imposed both strongly and weakly. It is shown that the weak and strong interface imposition produce similar continuous energy estimates. We discretize the problem in time and space by employing finite difference operators that satisfy a summation-by-parts rule. The interface and initial conditions are imposed weakly using a penalty formulation. It is shown that the results obtained for the weak interface conditions in the continuous case, lead directly to stability of the fully discrete problem.

We consider a stochastic analysis of non-linear viscous fluid flow problems with smooth and sharp gradients in stochastic space. As a representative example we consider the viscous Burgers’ equation and compare two typical intrusive and non-intrusive uncertainty quantification methods. The specific intrusive approach uses a combination of polynomial chaos and stochastic Galerkin projection. The specific non-intrusive method uses numerical integration by combining quadrature rules and the probability density functions of the prescribed uncertainties. The two methods are compared in terms of error in the estimated variance, computational efficiency and accuracy. This comparison, although not general, provide insight into uncertainty quantification of problems with a combination of sharp and smooth variations in stochastic space. It suggests that combining intrusive and non-intrusive methods could be advantageous.

The nonlinear incompressible Navier-Stokes equations with different types of boundary conditions at far fields and solid walls is considered. Two different formulations of boundary conditions are derived using the energy method. Both formulations are implemented in both strong and weak form and lead to an estimate of the velocity field. Equipped with energy bounding boundary conditions, the problem is approximated by using discrete derivative operators on summation-by-parts form and weak boundary and initial conditions. By mimicking the continuous analysis, the resulting semi-discrete as well as fully discrete scheme are shown to be provably stable, divergence free, and high-order accurate.

We consider a stochastic analysis of the non-linear viscous Burgers’ equation and focus on the comparison between intrusive and non-intrusive uncertainty quantification methods. The intrusive approach uses a combination of polynomial chaos and stochastic Galerkin projection. The non-intrusive method uses numerical integration by combining quadrature rules and the probability density functions of the prescribed uncertainties. The two methods are applied to a provably stable formulation of the viscous Burgers’ equation, and compared. As measures of comparison: variance size, computational efficiency and accuracy are used.

We present a modified formulation of the dual time-stepping technique which makes use of two derivatives in pseudo-time. This new technique retains and improves the convergence properties to the stationary solution. When compared with the conventional dual time-stepping, the method with two derivatives reduces the stiffness of the problem and requires fewer iterations for full convergence to steady-state. In the current formulation, these positive effects require that an approximation of the square root of the spatial operator is available and inexpensive. Funding agencies: Linkoping University; Swedish Governmental Agency for Innovation SystemsVinnova [2013-01209]; VINNOVAVinnova

Stochastic problems governed by nonlinear conservation laws are challenging due to solution discontinuities in stochastic and physical space. In this paper, we present a level set method to track discontinuities in stochastic space by solving a Hamilton-Jacobi equation. By introducing a speed function that vanishes at discontinuities, the iso-zeros of the level set problem coincide with the discontinuities of the conservation law. The level set problem is solved on a sequence of successively finer grids in stochastic space. The method is adaptive in the sense that costly evaluations of the conservation law of interest are only performed in the vicinity of the discontinuities during the refinement stage. In regions of stochastic space where the solution is smooth, a surrogate method replaces expensive evaluations of the conservation law. The proposed method is tested in conjunction with different sets of localized orthogonal basis functions on simplex elements, as well as frames based on piecewise polynomials conforming to the level set function. The performance of the proposed method is compared to existing adaptive multi-element generalized polynomial chaos methods. Funding agencies: Research Council of Norway through CLIMIT program [244035/E20 CONQUER]; U.S. Department of Energy Office of Science, Office of Advanced Scientific Computing Research [DE-SC0006402]; NSF [CMMI-1454601]

We study hyperbolic and incompletely parabolic systems with stochastic boundary and initial data. Estimates of the variance of the solution are presented both analytically and numerically. It is shown that one can reduce the variance for a given input, with a specific choice of boundary condition. The technique is applied to the Maxwell, Euler, and Navier–Stokes equations. Numerical calculations corroborate the theoretical conclusions.

We analyze the extension of summation-by-parts operators and weak boundary conditions for solving initial boundary value problems involving second derivatives in time. A wide formulation is obtained by first rewriting the problem on first order form. This formulation leads to optimally sharp fully discrete energy estimates that are unconditionally stable and high order accurate. Furthermore, it provides a natural way to impose mixed boundary conditions of Robin type, including time and space derivatives. We apply the new formulation to the wave equation and derive optimal fully discrete energy estimates for general Robin boundary conditions, including nonreflecting ones. The scheme utilizes wide stencil operators in time, whereas the spatial operators can have both wide and compact stencils. Numerical calculations verify the stability and accuracy of the method. We also include a detailed discussion on the added complications when using compact operators in time and give an example showing that an energy estimate cannot be obtained using a standard second order accurate compact stencil. Funding agencies: Swedish Research Council [621-2012-1689]

We make an initial investigation into the temporal efficiency of a fully discrete summation-by-parts approach for unsteady flows. As a model problem for the Navier–Stokes equations we consider a two-dimensional advection–diffusion problem with a boundary layer. The problem is discretized in space using finite difference approximations on summation-by-parts form together with weak boundary conditions, leading to optimal stability estimates. For the time integration part we consider various forms of high order summation-by-parts operators and compare with an existing popular fourth order diagonally implicit Runge–Kutta method. To solve the resulting fully discrete equation system, we employ a multi-grid scheme with dual time stepping.